If you already have a local user page on some wiki with content which you no longer want to be shown, you need to request its deletion. See Help:Deletion and undeletion for details, or contact an administrator of that wiki. Note that deleting the local user page will mean the user page will no longer appear in search results.

Where content comes from

Your global user page is generated on the central wiki, and the result is shown on local wikis. The content you see on the central wiki is what you'll see on other wikis. This means:

transcluding templates will use the central wiki's templates (your global user page won't use local templates);

transcluding special pages like {{Special:PrefixIndex/User:''My name''/}} will transclude the data from the central wiki;

the {{int:}}parser function will show system messages from the central wiki;

magic words like {{SERVERNAME}} will show the value from the central wiki.

Disabling your global user page

To disable the global user page, you can use the __NOGLOBAL__ magic word, which if present on the central user page, will prevent it from being displayed on remote wikis.
This will also determine whether a link to a user page on remote wiki should be red or blue.

Controlling what content is displayed

Just like transcluding a template, you can use the <noinclude> and <includeonly> tags to control what content is only displayed on the central site where the page exist, or globally on all sites.

You can “opt out” of global user pages by simply wrapping your central user page in <noinclude> tags. If the page is empty or consists only of invisible content like __NOINDEX__, it won't be shown on other wikis.

Caveats

Your user page might not look the same on other wikis due to differences in their site CSS. Check your page on a RTL wiki, e.g., :ar: or :he:.

Transclusions of special pages or parser functions like {{#time}} won't update as frequently as they do locally.

Page status indicators on the central wiki user page, e.g., an ICBM geohack in <indicator name="ICBM"><samp id="coordinates" class="coordinates plainlinks">…</samp></indicator>, will not work on local wikis. Try <noinclude><indicator name="ICBM"></noinclude><samp id="coordinates" class="coordinates plainlinks" style="float:right">…</samp><noinclude></indicator></noinclude> to get a page status indicator affecting only the user page on the central wiki, i.e., Meta-Wiki for most Wikimedia projects.